Finite Group Databasebeta

← #28346710

Subgroups — #28346710

Order 13696.

394 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20293235C211yesmaximal, characteristic, complemented
2#20293099C211yesmaximal, characteristic
3#20292971C211yesmaximal, characteristic, complemented
4#20292966C211yesmaximal, characteristic, complemented
5#20292929C211yesmaximal, characteristic, complemented
6#20292926C211yesmaximal, characteristic, complemented
7#20292831C211yesmaximal, characteristic, complemented
8#20292830C211yesmaximal, characteristic, complemented
9#20292779C211yesmaximal, characteristic
10#20292776C211yesmaximal, characteristic
11#20292727C211yesmaximal, characteristic, complemented
12#20292725C211yesmaximal, characteristic, complemented
13C4×#13133807C211yesmaximal, characteristic, complemented
14#20292686C211yesmaximal, characteristic, complemented
15C4×SD16×C107C211yesmaximal, characteristic, complemented
16C2×D4×D107C2211yescharacteristic
17C2×#13133807C2211yescharacteristic
18C2×#13133806C2211yescharacteristic
19C2×#13133805C2211yescharacteristic
20#16812107C2211yescharacteristic, complemented
21#16812104—21no—
22#16812103—21no—
23#16812101—21no—
24#16812100C2211yescharacteristic, complemented
25#16812099—21no—
26#16812089C2211yescharacteristic
27#16812085C2211yescharacteristic
28C4×Dic214C2211yescharacteristic, complemented
29C4×D428C2211yescharacteristic
30D4×Dic107C2211yescharacteristic
31#16812071C2211yescharacteristic
32C4×C22⋊D107—21no—
33(C22⋊D107)⋊C4—21no—
34#16812067C2211yescharacteristic
35#16812066C2211yescharacteristic
36C2×(C2×D107)⋊C4—21no—
37C22⋊C4×D107—21no—
38#16812042C2211yescharacteristic, complemented
39#16812042C2211yescharacteristic, complemented
40#16812041C2211yescharacteristic, complemented
41Dic107⋊C8C2211yescharacteristic, complemented
42Dic107⋊C8C2211yescharacteristic, complemented
43#16812034C2211yescharacteristic, complemented
44#16812033C2211yescharacteristic, complemented
45#16812032C2211yescharacteristic, complemented
46#16812031C2211yescharacteristic
47#16812030C2211yescharacteristic
48#16812029C2211yescharacteristic
49(C2×D107)⋊C8C2211yescharacteristic
50(C2×D107)⋊C8C2211yescharacteristic
51#16812020C2211yescharacteristic, complemented
52C2×SD16×C107C2211yescharacteristic
53C4×Q8×C107C2211yescharacteristic, complemented
54C4×D4×C107C2211yescharacteristic
55#106×C107C2211yescharacteristic, complemented
56#103×C107C2211yescharacteristic, complemented
57#102×C107C2211yescharacteristic
58C4×C8×C107C2211yescharacteristic, complemented
59C23×D107—21no—
60C2×Dic214C2311yescharacteristic
61C2×D428C2311yescharacteristic
62C2×C22⋊D107—21no—
63D4×D107—22no—
64D4×D107—22no—
65C22×Dic107—21no—
66C2×C4×D107—21no—
67C2×C4×D107—21no—
68C2×C4×D107C2311yescharacteristic
69#13133807—22no—
70#13133806—22no—
71#13133805—22no—
72C2×C107⋊C8C2311yescharacteristic
73C2×C107⋊C8C2311yescharacteristic
74C4⋊Dic107—21no—
75C4⋊Dic107—21no—
76C4⋊Dic107C2311yescharacteristic
77C4⋊Dic107C2311yescharacteristic
78Dic107⋊C4—21no—
79Dic107⋊C4—21no—
80Dic107⋊C4—21no—
81Dic107⋊C4—21no—
82Dic107⋊C4—21no—
83Dic107⋊C4D411yescharacteristic
84Dic107⋊C4D411yescharacteristic
85C22⋊Dic107—21no—
86(C2×D107)⋊C4—21no—
87(C2×D107)⋊C4—21no—
88(C2×D107)⋊C4—21no—
89(C2×D107)⋊C4—21no—
90(C2×D107)⋊C4—21no—
91(C2×D107)⋊C4D411yescharacteristic
92(C2×D107)⋊C4D411yescharacteristic
93C4×Dic107—21no—
94C4×Dic107—21no—
95C4×Dic107C2311yescharacteristic
96C2×Q8×C107C2311yescharacteristic
97C2×D4×C107C2311yescharacteristic
98C22×C4×C107—21no—
99SD16×C107—22no—
100C2×C8×C107C2311yescharacteristic
101C2×C8×C107C2311yescharacteristic
102C4⋊C4×C107—21no—
103C4⋊C4×C107C2311yescharacteristic
104C4⋊C4×C107C2311yescharacteristic
105C22⋊C4×C107—21no—
106C42×C107—21no—
107C42×C107C2311yescharacteristic
108C23×C107—21no—
109C22×D107—21no—
110C22×D107—24no—
111C22×D107—81no—
112C22×D107—81no—
113C22×D107C2×D411yescharacteristic
114Q8×C107—21no—
115Q8×C107—21no—
116D4×C107—21no—
117D4×C107#4112yes—
118C2×C4×C107—21no—
119C2×C4×C107—21no—
120C2×C4×C107—21no—
121C2×C4×C107—81no—
122C2×C4×C107C2×D411yescharacteristic
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2411yescharacteristic
125C22⋊D107—24no—
126C2×Dic107—21no—
127C2×Dic107—21no—
128C2×Dic107—21no—
129C2×Dic107—81no—
130C2×Dic107C2×D411yescharacteristic
131C2×Dic107C2×D411yescharacteristic
132C2×Dic107C2×D411yescharacteristic
133D428—21no—
134D428#4112yes—
135C4×D107—21no—
136C4×D107—21no—
137C4×D107—81no—
138C4×D107—81no—
139Dic214—21no—
140Dic214—21no—
141C8×C107—21no—
142C8×C107—21no—
143C107⋊C8—21no—
144C107⋊C8—21no—
145C22×C107—22no—
146C22×C107—41no—
147C22×C107—41no—
148C22×C107C22×D411yescharacteristic
149C2×D107—21no—
150C2×D107—22no—
151C2×D107—21no—
152C2×D107—21no—
153C2×D107—41no—
154C2×D107—41no—
155C2×D107—84no—
156C4×C107—21no—
157C4×C107—21no—
158C4×C107—41no—
159C4×C107—41no—
160C4×C107—41no—
161C4×C107C2×#4111yescharacteristic
162C4×C107C2×#4111yescharacteristic
163Dic107—21no—
164Dic107—21no—
165Dic107—21no—
166Dic107—41no—
167Dic107—41no—
168Dic107—41no—
169C2×C107—42no—
170C2×C107#52511yescharacteristic
171C2×C107#57711yescharacteristic
172C2×C107#57711yescharacteristic
173D107—41no—
174D107—42no—
175#3123—1071nomaximal, Sylow
176C107#312311yescharacteristic, complemented, Sylow
177#541—1071no—
178#518—1071no—
179#485—1071no—
180#485—1071no—
181#478—1071no—
182#478—1071no—
183#461—1071no—
184#461—1071no—
185#450—1071no—
186#450—1071no—
187#440—1071no—
188#440—1071no—
189C4×SD16—1071no—
190C4×SD16D10711yescharacteristic, complemented
191#433—1071no—
192C22×D4—1071no—
193C2×SD16—1071no—
194C2×SD16—1071no—
195C2×SD16—1071no—
196C2×SD16C2×D10711yescharacteristic
197C4⋊Q8—1071no—
198#126—2141no—
199#126—2141no—
200#125—1071no—
201#125—2141no—
202#125—2141no—
203C22⋊Q8—1071no—
204C22⋊Q8—1071no—
205C4×Q8—1071no—
206C4×Q8C2×D10711yescharacteristic, complemented
207C4×D4—1071no—
208C4×D4—1071no—
209C4×D4—1071no—
210C4×D4—2141no—
211C4×D4—2141no—
212C4×D4C2×D10711yescharacteristic
213#117—1071no—
214#117—1071no—
215C2×C22⋊C4—2141no—
216C2×C22⋊C4—2141no—
217#107—1071no—
218#107—1071no—
219#106—1071no—
220#106C2×D10711yescharacteristic, complemented
221C4⋊C8—1071no—
222C4⋊C8—1071no—
223#103—1071no—
224#103—1071no—
225#103—1071no—
226#103C2×D10711yescharacteristic, complemented
227#102—1071no—
228#102—1071no—
229#102—1071no—
230#102C2×D10711yescharacteristic
231C22⋊C8—1071no—
232C22⋊C8—1071no—
233#97—1071no—
234C4×C8C2×D10711yescharacteristic, complemented
235C24—2141no—
236C2×Q8—1071no—
237C2×Q8C22×D10711yescharacteristic
238C2×D4—1071no—
239C2×D4—2141no—
240C2×D4—2142no—
241C2×D4—2142no—
242C2×D4C22×D10711yescharacteristic
243C22×C4—21no—
244C22×C4—1071no—
245C22×C4—2141no—
246C22×C4—2141no—
247C22×C4—2141no—
248SD16—22no—
249SD16—2142no—
250SD16—2142no—
251SD16—2142no—
252C2×C8—1071no—
253C2×C8—1071no—
254C2×C8C22×D10711yescharacteristic
255C2×C8C22×D10711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—1071no—
258C4⋊C4—1071no—
259C4⋊C4—1071no—
260C4⋊C4—1071no—
261C4⋊C4—2141no—
262C4⋊C4—2141no—
263C4⋊C4—2141no—
264C4⋊C4—2141no—
265C4⋊C4—2141no—
266C4⋊C4—2141no—
267C4⋊C4—2141no—
268C4⋊C4C22×D10711yescharacteristic
269C4⋊C4C22×D10711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—1071no—
272C22⋊C4—1071no—
273C22⋊C4—2141no—
274C22⋊C4—2141no—
275C22⋊C4—2141no—
276C22⋊C4—2141no—
277C22⋊C4—2141no—
278C22⋊C4—2141no—
279C42—21no—
280C42—1071no—
281C42—2141no—
282C42—2141no—
283C42C22×D10711yescharacteristic
284C23—21no—
285C23—8561no—
286C23—8561no—
287C23—1071no—
288C23—2144no—
289C23—2141no—
290Q8—21no—
291Q8—21no—
292Q8—2141no—
293Q8—2141no—
294D4—21no—
295D4—1072no—
296D4—2144no—
297D4—2141no—
298D4#1313381812yes—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—8561no—
304C2×C4—8561no—
305C2×C4—8561no—
306C2×C4—1071no—
307C2×C4—1071no—
308C2×C4—1071no—
309C2×C4—2141no—
310C2×C4—2141no—
311C2×C4—2141no—
312C2×C4—2141no—
313C2×C4—2141no—
314C2×C4D4×D10711yescharacteristic
315C2×C4D4×D10711yescharacteristic
316C2×C4C23×D10711yescharacteristic
317C8—21no—
318C8—21no—
319C8—2141no—
320C8—2141no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—8564no—
325C22—4281no—
326C22—4281no—
327C22—2141no—
328C22—2142no—
329C22—2141no—
330C22—2141no—
331C22C2×D4×D10711yescharacteristic
332C4—21no—
333C4—21no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—4281no—
338C4—4281no—
339C4—4281no—
340C4—2141no—
341C4—2141no—
342C4—2141no—
343C4#41×D10711yescharacteristic
344C4C2×#1313381811yescharacteristic
345C2—42no—
346C2—4281no—
347C2—4282no—
348C2#2029313611yescharacteristic
349C2#2029343311yescharacteristic
350C2#2029343411yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radicalC2×D4×D107C22
hypercenterC4×SD16D107
hypocenterC107#3123
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20293235C2semidirect
3. #20292971C2semidirect
4. #20292966C2semidirect
5. #20292929C2semidirect
6. #20292926C2semidirect
7. #20292831C2semidirect
8. #20292830C2semidirect
11. #20292727C2semidirect
12. #20292725C2semidirect
13. C4×#13133807C2semidirect
14. #20292686C2semidirect
15. C4×SD16×C107C2semidirect
20. #16812107C22semidirect
24. #16812100C22semidirect
28. C4×Dic214C22semidirect
38. #16812042C22semidirect
39. #16812042C22semidirect
40. #16812041C22semidirect
41. Dic107⋊C8C22semidirect
42. Dic107⋊C8C22semidirect
43. #16812034C22semidirect
44. #16812033C22semidirect
45. #16812032C22semidirect
51. #16812020C22semidirect
53. C4×Q8×C107C22semidirect
55. #106×C107C22semidirect
56. #103×C107C22semidirect
58. C4×C8×C107C22semidirect
176. C107#3123semidirect
190. C4×SD16D107semidirect
206. C4×Q8C2×D107semidirect
220. #106C2×D107semidirect
226. #103C2×D107semidirect
234. C4×C8C2×D107semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202932352
#202930992
#202929712
#202929662
#202929292
#202929262
#202928312
#202928302
#202927792
#202927762
#202927272
#202927252
C4×#131338072
#202926862
C4×SD16×C1072
#3123107